2018/10/19 by Victor Goulart, Goulart, Victor, Nicolau C. Saldanha +1
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1810.08632
A parametric curve γ of class Cn on the n-sphere is said to be nondegenerate (or locally convex) when det(γ(t),γ'(t),⋯,γ(n)(t))>0 for all values of the parameter t. We orthogonalize this ordered basis to obtain the Frenet frame \mathfrakFγ of γ assuming values in the orthogonal group SOn+1 (or its universal double cover, Spinn+1), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve γ we assign its itinerary: a word w in the alphabet Sn+1\smallsetminus\e\ that encodes the succession of non open Schubert cells pierced by the complete flag of ℝn+1 spanned by the columns of \mathfrakFγ. Without loss of generality, we can focus on nondegenerate curves with initial and final flags both fixed at the (non oriented) standard complete flag. For such curves, given a word w, the subspace of curves following the itinerary w is a contractible globally collared topological submanifold of finite codimension. By a construction reminiscent of Poincaré duality, we define abstract cell complexes mapped into the original space of curves by weak homotopy equivalences. The gluing instructions come from a partial order in the set of words. The main aim of this construction is to attempt to determine the homotopy type of spaces of nondegenerate curves for n>2. The reader may want to contrast the present paper's combinatorial approach with the geometry-flavoured methods of previous works.